A DIFFERENTIAL VOLUMENOMETER, 


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ALBERT P* CARMAN. 


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. XXVI, No. 5, May, 1908] 


{ Reprinted from the PuysicAL Review, Vol 


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[Reprinted from the PHysICAL REVIEW, Vol. XXVI., No. 5, May, 1908. ] 


A DIFFERENTIAL VOLUMENOMETER. 
By ALBERT P, CARMAN. 


HE volumenometer or stereometer, invented by the French 
military engineer Say in 1797,’ remains in some of its forms, 

the only instrument for determining the volumes of small irregular 
bodies and powders which cannot be immersed in a liquid.. The 
fundamental principle of all volumenometers is the determination 
from Boyle’s law of the volume of air imprisoned in a space V by 


_ observing the change of pressure corresponding to a given change 


of volume. The volume is changed in all forms by raising and 
lowering mercury in a graduated tube, the upper part of which is 
the vessel 1. Then placing the body of volume x in the same space 
V, the new volume of air is found, and thus by subtraction, the 
volume x of the body is gotten. Most of the five or six volu- 
menometers? which have been described and used in scientific work, 
differ simply in the methods of raising and lowering the mercury, 
that is, in the means of changing the volumes and measuring the 
change. In the volumenometer of Oberdeck we have an instru- 
ment which is distinct from previous forms in being differential. It 
was from a study of Oberdeck’s instrument that the present work 
started. The first form described below was started simply as an 
improved form of Oberdeck’s instrument, but it was found to have 
some new features both in theory and manipulation, and this led 
to experiments with modifications, as will be described. 

Oberdeck’s form is shown in Fig. 1; V and V’ are two equal 


glass cups which can be closed by ground-glass plates; Z and 7” 


are graduated glass tubes fixed on a horizontal board and joined in 
aU at &. The U is connected by rubber tubing with a mercury 


1Say, Ann. de Chem. et Phys. (1), XXIII., 1797. 

2Regnault, Ann. de Chem. et Phys. (3), XIV., 1845. Kopp, Liebig’s Ann., 
XXXV., 1840. Ruedorf, Wied. Ann., VI., 1879. Paalzow, Wied. Ann., XIII., 
1881. Oberdeck, Wied. Ann., LXVII., 1899. Zehnder, Ann. d. Phys., X., 1903; 
XV., 1904. 


397 


A. P. CARMAN. [Vo1. XXVI. 


reservoir R. By raising and lowering A, the volume of air im- 
prisoned in V and V’’ can be changed. An essential feature of 
Oberdeck’s instrument is that the pressures on the two sides are 


Fig. si: 


always equal, since the two tubes are on the same level. The in- 


crements of volume are read by the position of the mercury columns 
in the graduated tubes 7. Some experiments showed that there 


Pigiez: 


must be considerable error in such 
readings owing to the lag of the 
mercury in horizontal tubes. It is 
also evident that the tubes 7 must 
be made long to get great sensitive- 
ness, since the device of enlarge- 
ments is not possible in horizontal 
tubes. To improve these features 
the instrument shown in Fig. 2 was 
made. A and # are vessels, each 
of about 5 c.c. capacity. A lid Z 
can be clamped air-tight on A. IZ 
is a glass tube connecting A and B. 
C is a graduated glass tube. This 
tube is connected by rubber tubing 
with a mercury reservoir Rk. | This 
reservoir can be raised and lowered 
so as to change the volume of the 
air imprisoned in AM/BC. On the 


right side we have A’J/’B’C’R’', a duplicate of the parts on the left 
side. The cups are connected by a sensitive manometer, containing 


No. 5.] A DIFFERENTIAL VOLUMENOMETER. 398 


a light aniline liquid. A stopcock X allows the ready equalizing 
of the pressures on the two sides. The whole is mounted on a verti- 
cal board ; on the same base is a vertical rod to which the platform 
carrying the mercury reservoirs is clamped. The reservoirs R and 
’ can be moved separately on rods carried by the common plat- 
form. In the first instrument made the vessels A and & were 
turned out of steel because they could be thus made exactly equal 
on the two sides. This equality is necessary in Oberdeck’s method. 
Making these vessels of steel involved sealing them to glass and 
this caused more trouble and delay than anything in these experi- 
ments. The volumenometer proved a very sensitive method of de- 
tecting air leaks. An instrument has now been made all of glass 
with mercury sealed joints. This has solved the leak trouble. 

The manipulation and theory are as follows: With the cups 4 
open to the air the mercury is raised to the marks /V/ and the ves- 
sels are then closed. Air is thus imprisoned at atmospheric pres- 
sure. The mercury reservoirs are lowered allowing the air volumes 
Vand V’ to expand by increments DV and DI” respectively. But 
during the operation the pressures on the two sides are kept equal 
by watching the manometer U. Letting p and 7’ represent the 
initial and final pressures, we have the relations 


Vp=(V4DV)p! and Vip=(V'4+DV')p". 


The mercury is next raised, restoring the volumes 1. A _ body of 
volume + is now placedin A. The mercury is again lowered, keep- 
ing the pressures equal on the two sides, and giving the air in A, the 
previous increment DV. This will correspond to an increment 
DV" on the right side for equality of pressure ~’’. We have then 


(V—x)p=(Ve—x44+ DV)p" and V'ip=(V'+ DV") p". 
From the above we have : 


VDV" — VIDYL 
.— DV" 


If the two volumes are equal, that is, V= V’’, then 


V(DV"—DV’) 
“es DV" 


399 A, P. CARMAN. [ VoL. XXVI. 


‘This is similar to Oberdeck’s formula. It is believed that the ver- 
tical tubes and the sensitive method of determining equality of 
pressure on the two sides are an advance. 

In working with the above volumenometer, an easier method of 
‘manipulation, using a simpler formula suggested itself. Starting as 
before, we have the relations 

Vp=(V+4DV)p Sand ip = Vay 
After introducing the body of volume %, we increase the volumes 
keeping the pressures equal on the two sides, but giving to V’ on 
the right side the previous increment DIV”’ and getting on the left 


the new increment DIV’. Evidently the pressure is the same JP’ as 
above. We thus get 


(V—xs)p=(V4+DV'—-x)p’ and V'p=(V'+Db"\p'. 
Substituting for Vp and transposing we get 

pl 
Poe 
The difference of volume increments (DV’ — DV /)is read off in 


terms of scale divisions of the tube C. The factor p’/( p — f’) can 
be determined directly from the pressures or it can be determined 


a 


(DV! — DV). 


experimentally by placing a body of known volume in A and thus 
calculating it. This calibration has of course to be done at least 
once for each series on account of temperature and pressure changes, 
and the results are then read off direct in scale divisions of the tube. 
It is to be noted that no assumptions are made here of an equality 
of volume of the two sides. In getting the volume of turned brass 
cylinder, which could be accurately calipered, the results were 
2.228 c.c. by the volumenometer and 2.210 c.c. by calipering. A 
steel ball was used for a standard. This shows the accuracy that 
has been attained ordinarily. 

The above experiments were made in the summer and fall of 
1906. In Sczence Abstracts for January, 1907, A. Lo Surdo’s dif- 
ferential volumenometer is described. This has something of the 
appearance of the differential form described above, but it is in fact 


quite different in construction and manipulation. 
UNIVERSITY OF ILLINOIS, ’ 
December, 1907. 


‘5 


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